Equipartition Theorem: Hamiltonian Form & Canonical Transformations

  • Thread starter Petar Mali
  • Start date
  • Tags
    Theorem
In summary, the conversation is about the Hamiltonian and whether the equipartition theorem is correct for this type of Hamiltonian. The speaker also mentions the possibility of a Hamiltonian that is not a function of squares of coordinates and impulses, and whether it can be obtained using canonical transformation. They also discuss the integrability and separability of the Hamiltonian, and the concept of ensemble averages. The conversation ends with a question about whether a "mysterious" Hamiltonian has the same form as the one mentioned earlier.
  • #1
Petar Mali
290
0
I have one question. If I have Hamiltonian:

[tex]H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2[/tex]

I can show that for the Hamiltonian of this type equipartition theorem is correct. Is there any Hamiltonian which is not a function od squares of coordinates and impulses are from which I can get this Hamiltonian some using canonical transformation. Example perhaps?
 
Physics news on Phys.org
  • #2
What does it mean "equipartition theorem is correct" ?
I think the hamiltonian is both integrable and separable, since it's not ergodic does the ensemble averages have sense?

Ll.
 
  • #3
You can prove that every degree of fredom have the same energy - equipartition theorem only for the Hamiltonian

[tex]
H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2
[/tex]

or maybe the Hamiltonian which canonical transformation is


[tex]
H=\sum^{f_1}_{i=1}\alpha_iq_i^2+\sum^{f_2}_{i=1}\beta_ip_i^2
[/tex]

I think that ''
mysterious '' Hamiltonian have the same form as one as I wrote! So for example


[tex]
K=\sum^{F_1}_{i=1}\alpha_iQ_i^2+\sum^{F_2}_{i=1}\beta_iP_i^2
[/tex]

Am I right?
 

1. What is the Equipartition Theorem?

The Equipartition Theorem is a fundamental principle in statistical mechanics that states that the total energy of a system is equally distributed among all of its degrees of freedom at thermal equilibrium.

2. What is the Hamiltonian form of the Equipartition Theorem?

The Hamiltonian form of the Equipartition Theorem is a mathematical representation of the principle that states that the energy of a system can be expressed as a sum of terms, each corresponding to a different degree of freedom.

3. What are canonical transformations in relation to the Equipartition Theorem?

Canonical transformations are mathematical transformations that preserve the form of the Hamiltonian in the Equipartition Theorem. They allow for the transformation of coordinates and momenta in a way that maintains the total energy of the system.

4. How does the Equipartition Theorem relate to the partition function?

The partition function, which is a fundamental concept in statistical mechanics, is directly related to the Equipartition Theorem. It is a mathematical function that describes the distribution of energies among the different states of a system, and it follows from the principle of equipartition.

5. What are some examples of systems that follow the Equipartition Theorem?

The Equipartition Theorem applies to a wide range of classical systems, including ideal gases, harmonic oscillators, and the motion of planets in a gravitational field. It also holds for more complex systems, such as solids and liquids, as long as they are at thermal equilibrium.

Similar threads

  • Classical Physics
Replies
2
Views
725
Replies
4
Views
561
Replies
5
Views
319
  • Classical Physics
Replies
7
Views
729
  • Classical Physics
Replies
1
Views
620
  • Classical Physics
Replies
6
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
653
Replies
3
Views
554
Replies
1
Views
931
  • Classical Physics
Replies
2
Views
897
Back
Top